I tried Cosmos via the Mongo compatibility layer and performance was abysmal, even when cranking the provisioned performance way up. I ended up getting significantly better performance from a reasonably sized MongoDB container running on AKS.
The former, lift and shift scenario I can understand as you need to structure your data to work best with CosmosDB/DynamoDB etc. If it's the latter that is useful to know as I'm currently building a POC on Azure so I should watch out for it.
This was all part of a benchmarking exercise to figure out how to scale an app with minimal changes so we expected some performance penalty, just not as steep as what we observed.
However, CosmosDB is a managed service, easier than setting up your own database. Did you exclude Atlas from your conclusion for some reason? Ie did AKS do things that Atlas could not?